Theorems · Definition · commutative algebra

LinearMap.compDer

{R : Type u_1} →
  {A : Type u_2} →
    {M : Type u_4} →
      [inst : CommSemiring R] →
        [inst_1 : CommSemiring A] →
          [inst_2 : AddCommMonoid M] →
            [inst_3 : Algebra R A] →
              [inst_4 : Module A M] →
                [inst_5 : Module R M] →
                  {N : Type u_5} →
                    [inst_6 : AddCommMonoid N] →
                      [inst_7 : Module A N] →
                        [inst_8 : Module R N] →
                          [inst_9 : IsScalarTower R A M] →
                            [inst_10 : IsScalarTower R A N] → (M →ₗ[A] N) → Derivation R A M →ₗ[A] Derivation R A N

We can push forward derivations using linear maps, i.e., the composition of a derivation with a linear map is a derivation. Furthermore, this operation is linear on the spaces of derivations.

Defined in

Mathlib.RingTheory.Derivation.Basic

Cited by

12 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringCommSemiringAddCommMonoidAlgebraModuleModuleAddCommMonoidModuleModuleIsScalarTowerIsScalarTower